Geometric aspects of polynomial interpolation in more variables

Transcription

1 Geometric aspects of polynomial interpolation in more variables BERLIN - 4/6/2010

2 Univariate polynomial interpolation A polynomial of degree at most d over a field K f () = a 0 + a a d d K[] depends on d + 1 parameters: its coefficients. Fi d + 1 distinct points and set the values 0,..., d A 1 K = K f ( i ) = f i K, i = 0,..., d. Then there is a unique such polynomial f (). Reason: linear algebra plus the fact that there is no non zero polynomial of degree d with zeros at 0,..., d.

3 Univariate Hermite interpolation Fi distinct points and positive integers 1,..., h A 1 K, m 1,..., m h N, m m h = d + 1 and set the values of the derivatives f (j 1) ( i ) = f i,j, i = 1,..., h, j = 1,..., m i Again there is a unique such polynomial f (), because there is no non zero polynomial of degree d with zeros of multiplicities at least m 1,..., m h at 1,..., h, i.e. Ruffini s theorem holds. Consequence: given a differentiable function F () of a real variable, we can uniquely approimate it with a polynomial of degree d by fiing d + 1 values of F () and of its derivatives. What is the situation for n 2 variables?

4 The multivariate case In general a polynomial f ( 0,..., n) K[ 0,..., n] of degree at most d depends on «d + n N n,d + 1 := n parameters, i.e. its coefficients. Fi points, positive integers and constants in K p i = ( i,1,..., i,n ) A n K = K n, m i > 0, f i,j K, i = 1,..., h, j = 1,..., m i with the condition and impose hx «mi + n 1 = N n,d + 1 n i=1 D (j 1) f (p i ) = f i,j, i = 1,..., h, j = 1,..., m i where D (k) is any derivative of order k. Is the resulting polynomial f uniquely determined? This is a linear system in the coefficients of f, whose associated homogeneous system is D (j 1) f (p i ) = 0, i = 1,..., h, j = 1,..., m i Is the only solution to this system the 0 polynomial? It is convenient to address this question in a more general, different, geometric setting.

5 Linear systems with multiple base points X is a projective, comple manifold of dimension n. L is a linear system of codimension one subvarieties, i.e. divisors, on X. p 1,..., p h are distinct points on X. m 1,..., m h are positive integers. L( P h i=1 m i p i ) L is the sublinear system formed by all divisors in L having multiplicity at least m i at the base points p i, i = 1,..., h, i.e. the local equation of the divisors in L( P h i=1 m i p i ) vanishes at p i with all its derivatives of order l m i 1. This imposes hx mi + n 1 i=1 n linear conditions on L.

6 Special systems The epected dimension of L( P h i=1 m i p i ) is: By linear algebra hx mi + n 1 e := ma{dim(l), 1} n i=1 hx hx dim(l( m i p i )) epdim(l( m i p i )) i=1 L( P h i=1 m i p i ) is said to be non special if i=1 i=1 hx hx dim(l( m i p i )) = epdim(l( m i p i )) it is called special otherwise: in this case the conditions imposed on L are dependent. i=1 Note: according to the definition, an empty system is non special.

7 The main question in this talk will be the... Problem (The dimensionality problem) Classify all special linear systems. Though more refined questions could be asked. However even this question is far too complicated! The answer depends on too many circumstances, e.g., it depends on the position of the points p 1,..., p h on X. Eample (An easy eample of special position of the points) If X = P r, and p 1,..., p h are on a line, they give dependent conditions to all hypersurfaces of degree d h 2. In any case, dim(l( P h i=1 m i p i )) is upper semicontinuous in the position of the points p 1,..., p h, hence it reaches its minimum for p 1,..., p h in sufficiently general position on X, i.e. for (p 1,..., p h ) in a suitable not empty Zariski open subset U m1,...,m h of X h.

8 The general dimensionality problem Take p 1,..., p h sufficiently general on X and set L( hx i=1 m i p i ) := L(m 1,..., m h ) = L(m l 1 1,..., m l t t ) The case t = 1 is called homogeneous. Define the general dimension of the system as hx gendim(l( m i p i )) := dim(l(m 1,..., m h )) i=1 Problem (The GDP) If p 1,..., p h are general, is gendim(l( P h i=1 m i p i )) equal to the epected dimension of L( P h i=1 m i p i )? If not then classify all systems L(m 1,..., m h ) which are special. The GDP is easy in the curve case (the answer is that no system with general base points is special in this case), but very complicated in general as soon as n = dim(x) 2. However a trivial situation is when m i = 1: general simple base points always impose independent conditions.

9 A wise reduction Given the compleity of the problem, it is wise to consider particular varieties X and linear systems L on them. Typically we take In this case where X = P n, L = L n,d := all degree d hypersurfaces. hx hx e := epdim(l n,d ( m i p i )) = ma{virtdim(l n,d ( m i p i )), 1} i=1 hx d + n hx mi + n 1 v := virtdim(l n,d ( m i p i )) = 1 n n i=1 i=1 is the so-called virtual dimension of the system. i=1 The GDP here coincides with the original polynomial interpolation problem. This is in general widely open, and there is even no conjectural answer to it. Ecept for the planar case...

10 The planar case p 1,..., p h general points in P 2, m 1,..., m h N multiplicities L = L d (m 1,..., m h ) = L d (m l 1 1,..., ml t t ) is the linear system of plane curves of degree d > 0 having multiplicity at least m i at p i for each i = 1,..., h. The virtual dimension of L is v := v(l) = d(d + 3)/2 X i m i (m i + 1)/2 The epected dimension is e := e(l) = ma{ 1, v} L is special if dim(l) > e(l)

11 Blow up One may formulate this on the blow up of P 2 at p 1,..., p h. The blow up is an algebraic surgery operation which substitutes to a point p in the plane (or on a surface) an eceptional curve E = P 1, with normal bundle of degree 1, called a (-1) curve. This is shown in red in the picture below:

12 Line bundles and cohomology X is the blow up of P 2 at p 1,..., p h, with the following divisor classes generating the Picard group Pic(X), i.e. the group of divisors modulo linear equivalence (which can be identified with the group of line bundles, i.e. vector bundles of rank 1, modulo isomorphism, on X): H the pull back of a line E 1,..., E h the eceptional divisors over p 1,..., p h The relevant line bundle on X is hx L = O X (dh m i E i ) i=1 and the virtual dimension is v = χ(l) 1 = h 0 (X, L) h 1 (X, L) 1. So L is non special if and only if i.e. L has natural cohomology. h 0 (X, L) h 1 (X, L) = 0

13 The Segre Harbourne Gimigliano Hirschowitz conjecture Naive conjecture: for general base points, L always has the epected dimension. This is wrong: e.g. look at L 2 (2 2 ) or to L 4 (2 5 ). ( 1) special systems: if L is not empty and C is a ( 1) curve on X then N L C = N < 0 = h 1 (X, L) 2 The two above eamples are of this type. a ( 1) curve on X is a curve C = P 1, with C 2 = 1, equivalently it can be blown down to a smooth point (Castelnuovo s theorem). Conjecture (SHGH) L is special if and only if it is ( 1) special. Equivalently, L is special if and only if the general curve in L has some multiple component (which turns out to be a fied ( 1) curve).

14 Nagata s conjecture Conjecture (Nagata, 1959) If h > 9, the base points are very general and L d (m 1,..., m h ) is not empty, then hx m i < d h i=1 Conjecture (Stronger Nagata s Conjecture (SNC)) If the h 0 base points are very general and L = L d (m 1,..., m h ) is not empty, not ( 1) special, then hx mi 2 i=1 d In particular, if C L is an irreducible curve different from a ( 1) curve, one has C 2 0 on the blow up. The SHGH Conjecture implies SNC, the converse does not hold. However Nagata s conjecture is an asymptotic form of the SHGH Conjecture. Nagata s original paper dealt with a negative answer to Hilbert s fourteenth problem. Nagata s conjecture arises also in other contets, like in symplectic packing problems (D. MacDuff L. Polterovich, P. Biran).

15 Nagata s conjecture and Seshadri constants (Chudnovski, D ly) Definition (MSC) Given p 1,..., p h P 2, the Multipoint Seshadri Constant is defined as ɛ(p 2 d, p 1,..., p h ) := inf{ P h i=1 m i hx : L d ( m i p i ), m i 0, i=1 hx m i > 0} i=1 If the points are very general denote it by ɛ h. One has ɛ(p 2, p 1,..., p h ) 1 h Nagata s conjecture asserts equality holds for very general points ɛ h = 1 h

16 Nagata s conjecture and the Mori cone If X is a comple, projective manifold, the Mori cone NE(X) of X is the closure of the conve cone spanned by the classes of effective curves inside N 1 (X), the R vector space dual to the space of R divisors modulo numerical equivalence. If X is the plane blown up at h points and l is the class of a line in N 1 (X) = Pic(X) R, consider the quadratic cone in N 1 (X) Q := {α : α 2 0, α l 0} Conjecture (Strong Nagata s Conjecture in Mori s setting) If X is the plane blown up at h very general points, then NE(X) = Q + X i E i where E i are the classes of ( 1) curves on X. The figure illustrates the case h 10. The denote the classes of ( 1) curves. The curved boundary of the Mori cone in the K + region has not yet been proved. K Q K + K K +

17 SHGH and Nagata in the homogeneous case For homogenous linear systems Nagata s and Strong Nagata s conjectures coincide. m d = mh d = m h N S d = m d 0 < h If for all (d, m) S the system L d (m h ) is non special, then for all (d, m) N the system L d (m h ) is empty; if for all (d, m) N the system L d (m h ) is empty, then for all (d, m) S the system L kd ((km) h ) is non special for k >> 0, actually S is contained in the ample cone.

18 Results (not comprehensive) SHGH holds for h 9 (Castelnuovo, 1891; Nagata, 1960; Gimigliano, Harbourne, 1986). SHGH holds for m i 11 (Dumnicki Jarnicki, 2005; Arbarello Cornalba, 1981: m i = 2; Hirschowitz, 1985: m i 3; Lorentz Lorentz, 1986; Mignon, 1998: m i 4; Yang, 2004: m i 7). SHGH holds for L d (m h ) for m 42 (Dumnicki, 2005; Ciliberto Miranda, 1998: m 12; Ciliberto Cioffi Miranda Orecchia, 2003: m 20). SHGH holds for L d (m h ) for h = k 2 points (Evain, 2005; Ciliberto Miranda, 2006; Roé, 2006; Nagata, 1960 proved Nagata Conjecture in this case). Hirschowitz and his followers (Gimigliano, Mignon, Evain, etc.) use a degeneration technique called the Horace method (i.e. divide et impera), consisting in eploiting subsequent specializations of the points on curves of (relatively) low degree. Ciliberto Miranda s approach is based on a different degeneration technique called the blow up and twist method, consisting in degenerating the plane together with the linear system.

19 The first interesting case h = 10 The virtual dimension of L d (m 10 ) is equal to 1, and one epects no such curves, for (d, m) in the following table: d m empty 3 1 easy: cubic through ten general points 19 6 posed by Dimier, solved by Hirschowitz early 80s Gimigliano s thesis le cas inviolé, according to A. Hirschowitz, see theorem below ? ? ?..? Theorem (Ciliberto Miranda, 2008) L d (m 10 ) has the epected dimension if d m In particular L 174(55 10 ) is empty. Theorem (Eckl, Ciliberto Dumitrescu Miranda Roé, 2008) ɛ

20 Projective toric varieties To give a pair (X, L), where: X is a projective, n dimensional toric variety an embedding X P r given by the sections of a line bundle L is equivalent to the datum of: an n dimensional integral compact conve polytope P R n +, determined up to integral affine isomorphisms. If consider the monomial map: P Z n = {m i = (m i1,..., m in ), 0 i r} φ P : (C ) n ( m 0 :... : mr ) P r where = ( 1,..., n) and m i = m i m in n The closure X P of the image of φ P is the image of X via the map determined by the sections of L.

21 Eample The d Veronese surface V 2,d in P d(d+3)/2 corresponds to the triangle: d = {(, y) : 0, y 0, + y d} d d It is the image of the plane via all monomials of degree d. For d = 1 we get the plane P 2 itself. For d = 2 we get the famous Veronese surface V 2,2 of degree 4 in P 5.

22 Eample P 1 P 1 embedded in P ab+a+b via all monomials of bidegree (a, b) in the variables ( 0, 1 ) and (y 0, y 1 ), corresponds to the rectangle: R a,b = {(, y) : 0 a, 0 y b} b For a = b = 1 we get four monomials X ij = i y j, 1 i j 2, verifying a unique quadratic relation i.e. we get a smooth quadric surface in P 3. a X 11 X 22 X 12 X 21 = 0

23 Toric degenerations (I) Consider a subdivision D of the polytope P defining the toric variety X P of dimension n, i.e., a finite family of n dimensional conve polytopes whose union is P and any two of them intersect only along a common face. D is called regular if there is a piecewise linear, positive function F defined on P such that: (i) the polytopes of D are the orthogonal projections on the hyperplane z = 0 of R n+1 of the n dimensional faces of the graph polytope G(F ) := {(, z) P R : 0 z F ()} which are neither vertical, nor equal to P; (ii) the function F is strictly conve i.e., the hyperplanes determined by each face of G(F ) intersect G(F ) only along that face.

24 Eample (A simple eample of a regular subdivision)

25 Toric degenerations (II) Given a regular subdivison D, there is a one parameter, flat degeneration of X P, to a reducible toric variety X 0 = [ X Q Q D If Q and Q have a common face R, then X Q X Q = X R Description of the degeneration: φ D : (, t) (C ) n C (t F(m 0) m 0 :... : t F (mr ) mr ) P r X t = closure of the image of φ D (, t), for t 0, is a copy of X P. X 0 is the limit of X t when t tends to 0.

26 Eample (The quadric degeneration of V 2,d ) The regular subdivision illustrated below for d = 6 gives a degeneration of V 2,d to a union of d planes and `d 2 quadrics. A corresponding strictly conve piecewise linear function is determined by the conditions F (i, j) = i 2 + j 2 for i, j non negative integers. The vertices of this configuration of planes and quadrics are linearly independent in the ambient P d(d+3)/2 and can be taken as the coordinate points.

27 Eample (Planar degenerations of V 2,d ) Each quadric can independently degenerate to a union of two planes, in two possible ways: or for each quadric. This way one finds several different degenerations of the Veronese into a union of planes, i.e. a a planar degeneration.

28 Tropical geometry There are connections with tropical geometry here: toric degenerations of V 2,d are closely related (essentially equivalent, indeed) to plane tropical curves of degree d. E.g., we see below a subdivison of 2 corresponding to a planar degeneration of the Veronese surface V 2,2 in P 5 and a related tropical conic (in green).

29 Double point interpolation This is the only general result in interpolation in more than two variables: Theorem (Aleander Hirschowitz, 1996) L n,d (2 h ) is non special unless n any d h 2,..., n The original, complicated proof used the Horace method. Simplifications, with the same technique, by K. Chandler (2002), Brambilla Ottaviani (2008). The blow up and twist method has been applied by E. Postinghel (2010) in her thesis. There has been recent activity in trying to find a combinatorial (tropical) proof of this theorem using the above techniques: Draisma (2004), Ciliberto Dumitrescu Miranda (2007), for n = 2; Brannetti (2007), for n = 3. Similar ideas also work for other toric varieties (Draisma, 2007).

30 An idea of the combinatorial proof In case n = 2, one uses the following: Lemma (The basic combinatorial lemma) Suppose there is a planar degeneration D of V 2,d, and a set of h pairwise disjoint planes of D. Then L d (2 h ) has the epected dimension e = d(d + 3) 2 3h Sketch of the proof of Aleander Hirschowitz theorem for n = 2: verify that L d (2 h ) has the epected dimension whenever d 5 and h = (d + 1)(d + 2)/6 With this number of points, the virtual dimension of L d (2 h ) is ( 1 if d 1, 2 mod 3 v = d(d + 3)/2 3h = 0 if d 0 mod 3.

31 Do by hands the cases 5 d 10, using the basic combinatorial lemma, e.g. D 5 : D 6 :

32 Then proceed by induction, assuming the result holds for d 6. Make a planar degeneration of V 2,d, starting with: D d 6 strip of height 1 strip of height 5 By induction, we know how to subdivide D d 6 in order to get the maimal number of pairwise disjoint triangles. Triangulate the central strip of height 1 as you like, and take no triangles there.

33 It remains to triangulate the lower strip of height 5 and choose the pairwise disjoint triangles there. The strip contains (d + 1)/2 3 copies of the rectangle:

34 On the far right complete using the configurations D 5 and D 6 presented above: D d 6 D d 6 d odd D 5 d even D 6 Brannetti s proof for r = 3 is in the same style.

35 Secant varieties Why does the basic combinatorial lemma hold? The reason is geometric, related to secant varieties. X P r a projective variety of dimension n, spanning P r. S k (X) = is the k secant variety of X. One has [ p 0,...,p k X,p 0,...,p k l.i. p 0,..., p k epdim(s k (X)) = min{r, (k + 1)(n + 1) 1} dim(s k (X)) X is called k defective if strict inequality holds. By Terracini s lemma, X is k defective if and only if L(2 k+1 ) is special, with L = the linear system of hyperplane sections of X. Secant varieties are an authentic crossroad in mathematics (and not only!): besides their interest in algebraic geometry, secant varieties arise in a number of other fields, like algebra, representation theory, projective differential geometry, topology, stochastics and algebraic statistics. Recently they became particularly useful in biology, especially in phylogenetics.

36 An eample of applications to algebra: the Waring s problem Aleander Hirschowitz Theorem provides the list of k defective Veronese varieties V n,d P N n,d, N n,d = `n+d n 1. This answer the so called Waring s problem for forms. Fi positive integers d, k, n. When may we write a form f ( 0,..., n) of degree d as a sum of k + 1 d th powers of linear forms l i ( 0,..., n), i = 0,..., k, i.e. as f ( 0,..., n) = kx l i ( 0,..., n) d? i=0 If this happens, we say that the k Waring property holds for f. V d,n can be seen as the proportionality classes of non zero forms of type l( 0,..., n) d with l( 0,..., n) linear. Then the k Waring property holds for f if and only if [f ] S k (V n,d ). The Waring problem has its roots in number theory: given positive integers d, h, may we write any positive integer as a sum of h non-negative d th powers? E.g., for d = 2 and h = 4, this is affirmatively answered by the celebrated Gauss Theorem.

37 Strict Waring property and ranks We say that the strict (n, d, k) Waring property holds for a general [f ] S k (V n,d ), if the epression kx f ( 0,..., n) = l i ( 0,..., n) d i=0 with l i linear forms, is unique up to multiplication by a constant. This what biologist call the identifiability condition. It is equivalent to the geometric condition: the general point in S k (V n,d ) sits in a unique k dimensional subspace which is (k + 1) secant to V n,d. Strict Waring property provides a canonical form for forms enjoying it. In general, Waring property gives notions of rank for forms, similar to the rank of tensors which are useful in numerical analysis: f has rank k + 1 if k is the minimum such that kx f ( 0,..., n) = l i ( 0,..., n) d i=0 and has border rank k + 1 if k is the minimum such that [f ] S k (V n,d ). In general the border rank is smaller than the rank.

38 Degenerations and secant varieties Problem Let D be a planar degeneration of a toric surface X. What is the limit of S k (X)? (Similar questions can be asked for higher dimensional toric varieties.) References: Sturmfels Sullivant (delightful degenerations), Co Sidman, Ciliberto Dumitrescu Miranda. A remark: if there is a (k + 1) tuple of independent planes in D, then they span a linear space of dimension 3k + 2 sitting in the limit of S k (X), which is therefore not k defective. This proves the Basic Combinatorial Lemma. A speculation: if S k (X) has the epected dimension 3k + 2, then the limit of S k (X) is the union of all (3k + 2) subspaces spanned by (k + 1) tuples of independent planes in D. If this happens, we say the degeneration is almost k delightful. The eistence of almost delightful degenerations helps in computing the degree of S k (X), which is a hard problem, unsolved in general.

39 Hilbert s Theorem revisited Theorem (Hilbert s Theorem; see also Ein Sheperd-Barron) There is a unique 6 space passing through a general point of P 20 and 7 secant the Veronese surface V 2,5, i.e. the strict Waring property holds in this case, giving a canonical form for the general quintic homogeneous polynomial in three variables. Indeed, the configuration of 7 independent planes in D 5 shown below is unique: D 5 : There is work in progress on these ideas about: higher multiplicities, higher dimension, influence of the singularities of the degeneration on calculations of the degrees of secant varieties, etc.

40 Conclusions I tried to show how interpolation, originated from elementary analysis and algebra, has deep algebro geometric aspects as well as applications to other seemingly distant mathematical fields, e.g. symplectic geometry. Inside algebraic geometry, we see relations with the projective geometry of secant varieties, which in turn applies again to algebra via tensor rank computation, Waring problem, canonical forms, enumerative problems, etc. They have also recent striking applications to natural science, e.g. to phylogenetics. Various techniques are used in this field, among others: degeneration techniques; toric and tropical geometry; combinatorial techniques. The field is active and most of the basic, deepest problems like Segre Harbourne Gimigliano Hirschowitz conjecture, Nagata s conjecture, etc. are widely open.